Controlled peeling of fully convex digital sets and thin envelope

Full convexity is an interesting alternative to classical digital convexity since it guarantees connectedness and even simple connectedness in digital spaces Zd, for any dimension d. This paper aims at giving a better understanding of the monotonicity properties of fully convex digital sets, since earlier works showed that the question was difficult for thin fully convex sets. To decipher the hierarchy of fully convex sets ordered by inclusion, we study how we can peel a fully convex set progressively while keeping its full convexity. We provide a characterization of peelable points and fast algorithms to identify them. Furthermore we show that fully convex set can be peeled one point at a time till reduced to the empty set, or to any fully convex subsets, similarly to digitally convex sets in the classical sense. The peeling of a fully convex set can be seen as an analog to homotopic thinning processes, but with an additional geometric property. Finally, we propose a new envelope operator that builds a fully convex set from an arbitrary digital set, by dilation then peeling. This new operator is shown to be superior to former envelope operators, because it stays at bounded distance of the Euclidean convex hull.